# 1-forms on Riemann surface with boundary

Suppose $R$ is orientable compact Riemann surface with boundary $\partial R$ a collection $C_1, \dots C_k$ of oriented circles, so that for $i=1,\dots,j$, $C_i$ has boundary orientation, and for $i=j+1,\dots k$, $C_i$ has nonboundary orientation.

Suppose moreover that have parametrized neighborhoods $U_i$ of $C_i$ in $R$ so that $U_i$ is halfcylinder, so that for $i=1,\dots j$, then $U_i$ is $]-\infty,0] \times \mathbf{S}^1$, and for $i=j+1,\dots k$, then $U_i$ is $[0,\infty[\times \mathbf{S}^1$.

Pick complex coordinate $s+it$ on each $U_i$, pick numbers $a_i$ such that

$$a_1 + \dots + a_j - a_{j+1} - \dots - a_k = 0.$$

This condition implies that it is possible to find 1-form $\omega$ on $R$ so that $d \omega=0$ and so that on $U_i$ the 1-form $\omega$ is constant 1-form $a_i dt$.

Question For given fixed set $(a_1,\dots a_k)$ of numbers satisfying above condition, let $\Omega(a_1,\dots,a_k)$ denote space of all such 1-forms $\omega$. Is it possible to give nice description (structure of manifold) of $\Omega(a_1,\dots a_k)$?

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Well the one-forms are characterized by the tuples $(a_1,\dots,a_k)$ with the condition that $(a_1 + \cdots + a_j) - (a_{j+1}+\cdots + a_k) = 0$. So if we define $f: \mathbb{C}^j\times \mathbb{C}^{k - j}\rightarrow\mathbb{C}$ by $f(u, v) = (u_1 + \cdots + u_j) - (v_1 + \cdots + v_{k - j})$ and set $M = f^{-1}(0)$, what can you conclude about $M$? – William Apr 6 '12 at 10:46

It's just the space of smooth functions on $R$ that vanish on $\partial R$. Which is a Fréchet manifold at least...